{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/higher-order-stochastic-integration-through","title":"Higher-order stochastic integration through cubic stratification","arxiv_id":"2210.01554","date":"2022-10-04","proceeding":null,"authors":["Nicolas Chopin","Mathieu Gerber"],"abstract":"We propose two novel unbiased estimators of the integral $\\int_{[0,1]^{s}}f(u) du$ for a function $f$, which depend on a smoothness parameter $r\\in\\mathbb{N}$. The first estimator integrates exactly the polynomials of degrees $p<r$ and achieves the optimal error $n^{-1/2-r/s}$ (where $n$ is the number of evaluations of $f$) when $f$ is $r$ times continuously differentiable. The second estimator is computationally cheaper but it is restricted to functions that vanish on the boundary of $[0,1]^s$. The construction of the two estimators relies on a combination of cubic stratification and control ariates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of $n$.","url_abs":"https://arxiv.org/abs/2210.01554v2","url_pdf":"https://arxiv.org/pdf/2210.01554v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"higher-order-stochastic-integration-through","repo_url":"https://github.com/nchopin/cubic_strat","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2210.01554","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}